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  <div class="section" id="numpy-random-randomstate-multivariate-normal">
<h1>numpy.random.RandomState.multivariate_normal<a class="headerlink" href="#numpy-random-randomstate-multivariate-normal" title="Permalink to this headline">¶</a></h1>
<p>method</p>
<dl class="method">
<dt id="numpy.random.RandomState.multivariate_normal">
<code class="sig-prename descclassname">RandomState.</code><code class="sig-name descname">multivariate_normal</code><span class="sig-paren">(</span><em class="sig-param">mean</em>, <em class="sig-param">cov</em>, <em class="sig-param">size=None</em>, <em class="sig-param">check_valid='warn'</em>, <em class="sig-param">tol=1e-8</em><span class="sig-paren">)</span><a class="headerlink" href="#numpy.random.RandomState.multivariate_normal" title="Permalink to this definition">¶</a></dt>
<dd><p>Draw random samples from a multivariate normal distribution.</p>
<p>The multivariate normal, multinormal or Gaussian distribution is a
generalization of the one-dimensional normal distribution to higher
dimensions.  Such a distribution is specified by its mean and
covariance matrix.  These parameters are analogous to the mean
(average or “center”) and variance (standard deviation, or “width,”
squared) of the one-dimensional normal distribution.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>New code should use the <code class="docutils literal notranslate"><span class="pre">multivariate_normal</span></code> method of a <code class="docutils literal notranslate"><span class="pre">default_rng()</span></code>
instance instead; see <em class="xref py py-obj">random-quick-start</em>.</p>
</div>
<dl class="field-list">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>mean</strong><span class="classifier">1-D array_like, of length N</span></dt><dd><p>Mean of the N-dimensional distribution.</p>
</dd>
<dt><strong>cov</strong><span class="classifier">2-D array_like, of shape (N, N)</span></dt><dd><p>Covariance matrix of the distribution. It must be symmetric and
positive-semidefinite for proper sampling.</p>
</dd>
<dt><strong>size</strong><span class="classifier">int or tuple of ints, optional</span></dt><dd><p>Given a shape of, for example, <code class="docutils literal notranslate"><span class="pre">(m,n,k)</span></code>, <code class="docutils literal notranslate"><span class="pre">m*n*k</span></code> samples are
generated, and packed in an <em class="xref py py-obj">m</em>-by-<em class="xref py py-obj">n</em>-by-<em class="xref py py-obj">k</em> arrangement.  Because
each sample is <em class="xref py py-obj">N</em>-dimensional, the output shape is <code class="docutils literal notranslate"><span class="pre">(m,n,k,N)</span></code>.
If no shape is specified, a single (<em class="xref py py-obj">N</em>-D) sample is returned.</p>
</dd>
<dt><strong>check_valid</strong><span class="classifier">{ ‘warn’, ‘raise’, ‘ignore’ }, optional</span></dt><dd><p>Behavior when the covariance matrix is not positive semidefinite.</p>
</dd>
<dt><strong>tol</strong><span class="classifier">float, optional</span></dt><dd><p>Tolerance when checking the singular values in covariance matrix.
cov is cast to double before the check.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl>
<dt><strong>out</strong><span class="classifier">ndarray</span></dt><dd><p>The drawn samples, of shape <em>size</em>, if that was provided.  If not,
the shape is <code class="docutils literal notranslate"><span class="pre">(N,)</span></code>.</p>
<p>In other words, each entry <code class="docutils literal notranslate"><span class="pre">out[i,j,...,:]</span></code> is an N-dimensional
value drawn from the distribution.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="numpy.random.Generator.multivariate_normal.html#numpy.random.Generator.multivariate_normal" title="numpy.random.Generator.multivariate_normal"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Generator.multivariate_normal</span></code></a></dt><dd><p>which should be used for new code.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>The mean is a coordinate in N-dimensional space, which represents the
location where samples are most likely to be generated.  This is
analogous to the peak of the bell curve for the one-dimensional or
univariate normal distribution.</p>
<p>Covariance indicates the level to which two variables vary together.
From the multivariate normal distribution, we draw N-dimensional
samples, <img class="math" src="../../../_images/math/7e7cbe004b0d97071d449df48c47de3cf31a5609.svg" alt="X = [x_1, x_2, ... x_N]"/>.  The covariance matrix
element <img class="math" src="../../../_images/math/6b70bcae52a568a124c4c8024147cabbfb28ce2f.svg" alt="C_{ij}"/> is the covariance of <img class="math" src="../../../_images/math/c67734af70861b2bd4dedf5c41c9aad231466f84.svg" alt="x_i"/> and <img class="math" src="../../../_images/math/ab9afdaf786ce53318d75d81f050af8560822fcd.svg" alt="x_j"/>.
The element <img class="math" src="../../../_images/math/e0230477f2ec2f4d92596220cec9555ca8d99c84.svg" alt="C_{ii}"/> is the variance of <img class="math" src="../../../_images/math/c67734af70861b2bd4dedf5c41c9aad231466f84.svg" alt="x_i"/> (i.e. its
“spread”).</p>
<p>Instead of specifying the full covariance matrix, popular
approximations include:</p>
<blockquote>
<div><ul class="simple">
<li><p>Spherical covariance (<em class="xref py py-obj">cov</em> is a multiple of the identity matrix)</p></li>
<li><p>Diagonal covariance (<em class="xref py py-obj">cov</em> has non-negative elements, and only on
the diagonal)</p></li>
</ul>
</div></blockquote>
<p>This geometrical property can be seen in two dimensions by plotting
generated data-points:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">mean</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">cov</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">100</span><span class="p">]]</span>  <span class="c1"># diagonal covariance</span>
</pre></div>
</div>
<p>Diagonal covariance means that points are oriented along x or y-axis:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">multivariate_normal</span><span class="p">(</span><span class="n">mean</span><span class="p">,</span> <span class="n">cov</span><span class="p">,</span> <span class="mi">5000</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="s1">&#39;x&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;equal&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p>Note that the covariance matrix must be positive semidefinite (a.k.a.
nonnegative-definite). Otherwise, the behavior of this method is
undefined and backwards compatibility is not guaranteed.</p>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r4da0e147c95f-1"><span class="brackets">1</span></dt>
<dd><p>Papoulis, A., “Probability, Random Variables, and Stochastic
Processes,” 3rd ed., New York: McGraw-Hill, 1991.</p>
</dd>
<dt class="label" id="r4da0e147c95f-2"><span class="brackets">2</span></dt>
<dd><p>Duda, R. O., Hart, P. E., and Stork, D. G., “Pattern
Classification,” 2nd ed., New York: Wiley, 2001.</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">mean</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">cov</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">multivariate_normal</span><span class="p">(</span><span class="n">mean</span><span class="p">,</span> <span class="n">cov</span><span class="p">,</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(3, 3, 2)</span>
</pre></div>
</div>
<p>The following is probably true, given that 0.6 is roughly twice the
standard deviation:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">-</span> <span class="n">mean</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mf">0.6</span><span class="p">)</span>
<span class="go">[True, True] # random</span>
</pre></div>
</div>
</dd></dl>

</div>


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